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|Title||A Family of Cyclic Codes Over Finite Chain Rings|
Codes over flnite rings have received much attention recently after it was proved that important families of binary non-linear codes are images under a Gray map of linear codes over Z4. A set of n-tuples over a ring R is called a code over R if it is an R-module. A cyclic codes of length n over the ring R is a linear code with property that if the codeword (c0; c1; … ; cn¡1) ∈ C then the cyclic shift (c1; c2; … ; c0) ∈ C. The cyclic codes are ideals in the ring R n = R[x]=(xn \ 1). A commutative ring R with identity 1 ≠ 0 is called a flnite chain ring if its ideals are linearly ordered by inclusion. We study in this thesis to study cyclic codes over flnite chain rings. We flrst give a survey study a bout cyclic codes over the rings Zpk of integers modulo pk for a prime p and k ‚ 1; in particular Z4 and Z8 and study their structures. We will extend this study to cyclic codes for more chain rings Fp + uFp, Fp + uFp + u2Fp and F p + uFp + … + uk¡1Fp for difierent prime number p and we will deflne and construct idempotent generators for cyclic over these rings and study their properties.
|Publisher||the islamic university|
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